Exercise 22, § 7

Assume that $\mathfrak{g}$ is simple, in other words that R is irreducible. Show that there exists a unique dominant weight $\lambda \not= 0$ such that the set of weights of $E(\lambda )$ is $W.\lambda \cup \{0\}:$ we have $\lambda =\alpha$, where $\alpha \in R$ is such that $H_{\alpha}$ is the highest root of $R^{\vee}$. When all the roots are of the same length (cases $A_l,D_l,E_6,E_7,E_8$), we have $\lambda = \widetilde{\alpha}$; this is the only root that is a dominant

$^{15}$For more details, cf. E. B. DYNKIN, Maximal subgroups of classical groups [in

Russian], Trudy Moskov. Mat. Ob$\check{s}\check{c}$., Vol. 1 (1952), pp. 39-166 (= Amer. Math.

Soc. Transl., Vol. 6 (1957), pp. 245-374). weight; the corresponding representation is the adjoint representation of $\mathfrak{g}$. In the other cases, $\lambda$ is the only root of minimum length that is a dominant weight; with the notations of Chap. VI, Plates, we have: $\lambda =\varpi_1$ (type $B_l$); $\lambda =\varpi_2$ (type $C_l$)$;\lambda =\varpi_4$ (type $F_4$)$;\lambda =\varpi_1$ (type $G_2$).